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In this note we construct an algorithm generating any discrete distribution with an arbitrary coin (and, as a result, with arbitrary initial distribution). The coin need not be fair and the target distribution can be supported on a countable set.

We study three properties associated to the recurrence of orbits in flows: asymptotic periodicity, positive asymptotic periodicity and G-asymptotic periodicity. We determine which implications between these notions hold and which do not. We also show how these notions are related to Lyapunov stability.

We introduce the asymptotic property associated with recurrence-like behavior of orbits in dynamical systems in general metric spaces. We define a notion of weak asymptotic periodicity and determine its elementary properties and relations including the invariance by topological conjugacy. We use the equicontinuity and the topology of the space to describe necessary and sufficient conditions for the existence of such a behavior.

In this article we describe selected topological and dynamical properties of asymptotically periodic motions in continuous dynamical systems (flows). The main result is to show Lagrange stability (i.e. the closure of positive orbit is compact) of such motion with the aid of topological properties of limit sets. Two sufficient conditions for this kind of stability are provided: the value of asymptotic period (our approach to this notion differs from the usual one) and proper metric space property.